Pharmacokinetics – One-Compartment Open Model
This is the mathematical core of the subject. It introduces pharmacokinetic modeling — compartment models, non-compartment analysis, and physiological models. The one-compartment open model is studied in detail for three scenarios: IV bolus (instantaneous input), IV infusion (constant-rate input), and extravascular/oral administration (first-order absorption). Key pharmacokinetic parameters (KE, t½, Vd, AUC, Ka, CLt, CLR) are defined, calculated, and their clinical significance explained.
Syllabus & Topics
- 1Pharmacokinetics – Definition: Pharmacokinetics (PK) = what the body does to the drug. Quantitative study of the time course of drug absorption, distribution, metabolism, and excretion (ADME). Expressed through mathematical equations relating plasma concentration to time. Applications: dose calculation, dosage regimen design, therapeutic drug monitoring, bioequivalence studies, drug-drug interaction prediction. Pharmacodynamics (PD) = what the drug does to the body (effect). PK/PD together: link dose → concentration → effect.
- 2Types of PK Models: (1) Compartment models: body = one or more hypothetical compartments connected by rate constants. Mammillary model (central + peripheral compartments). Catenary model (compartments in series — rarely used). (2) Non-compartment (model-independent) analysis: uses statistical moments (AUC, AUMC, MRT) — no assumption about number of compartments. (3) Physiological (PBPK) models: each organ = separate compartment with real anatomical volumes and blood flows. Most realistic but complex. Compartment models are most commonly used in pharmacokinetics courses.
- 3One-Compartment Open Model – Assumptions: (1) Body = single, well-mixed compartment. (2) Drug distributes INSTANTANEOUSLY and uniformly throughout the body. (3) ‘Open’ = drug can enter and leave the compartment. (4) Elimination follows first-order kinetics: dCp/dt = −KE·Cp. Best suited for: drugs that distribute rapidly (small Vd) relative to elimination. Limitations: does not describe distribution phase (no α-phase). For drugs with significant distribution lag, use two-compartment model.
- 4One-Compartment IV Bolus: Drug injected directly into blood → C₀ is maximum immediately. Cp(t) = C₀ · e^(-KE·t). Semi-log plot: ln(Cp) = ln(C₀) − KE·t → straight line with slope = −KE. Parameters from the plot: KE (slope), C₀ (y-intercept), t½ = 0.693/KE, Vd = Dose/C₀, AUC = C₀/KE = Dose/(KE·Vd) = Dose/CL, CL = KE·Vd. Example: 500 mg IV bolus, C₀ = 10 mg/L → Vd = 500/10 = 50 L. If KE = 0.1 h⁻¹ → t½ = 6.93 h, CL = 0.1 × 50 = 5 L/h, AUC = 10/0.1 = 100 mg·h/L.
- 5One-Compartment IV Infusion: Drug infused at constant rate R₀ (mg/h). Cp rises gradually → reaches plateau (steady state, Css). Cp(t) = (R₀/CL) × (1 − e^(-KE·t)). At steady state (t → ∞): Css = R₀/CL = R₀/(KE·Vd). Time to reach steady state: depends ONLY on t½ — not on infusion rate. ~90% Css reached in 3.32 × t½; ~97% in 5 × t½. Rule of thumb: steady state takes 4-5 half-lives. To reach target Css faster → give loading dose (IV bolus) followed by infusion. Loading dose = Css × Vd = R₀/KE.
- 6One-Compartment Extravascular (Oral): Drug absorbed from GI tract with first-order rate constant Ka, then eliminated with KE. Cp(t) = (F·Ka·Dose)/(Vd·(Ka−KE)) × (e^(-KE·t) − e^(-Ka·t)). Where F = bioavailability fraction. Plasma profile: absorption phase (Cp rising) → peak (Cmax at Tmax) → elimination phase (Cp declining). At Tmax: rate of absorption = rate of elimination (dCp/dt = 0). Tmax = ln(Ka/KE) / (Ka − KE). Method of Residuals (Feathering/Peeling): separate Ka and KE from oral data by subtracting the terminal elimination line from observed data → residual line gives Ka.
- 7Elimination Rate Constant (KE): First-order rate constant for drug elimination. Units: time⁻¹ (h⁻¹). KE = sum of all elimination rate constants: KE = Km + KR + Kb + … (metabolism, renal, biliary, etc.). Determined from: slope of terminal portion of ln(Cp) vs time plot (IV bolus: slope = −KE; oral: terminal slope = −KE). Significance: reflects overall rate of drug removal from body. Large KE → fast elimination → short t½; Small KE → slow elimination → long t½.
- 8Half-Life (t½): Time for plasma concentration to decrease by 50%. t½ = 0.693/KE = (0.693 × Vd)/CL. Determined from: semi-log plot (time for Cp to halve). Significance: (1) Determines dosing interval (typically dose every t½ for IV, or 1-2 × t½ for oral). (2) Time to reach steady state (4-5 × t½). (3) Time for drug to be >97% eliminated after stopping (5 × t½). (4) Drug accumulation: longer t½ → more accumulation with repeated dosing. Half-life depends on BOTH Vd and CL — if Vd ↑ or CL ↓, t½ increases.
- 9Volume of Distribution (Vd): Vd = Dose_IV / C₀. Hypothetical volume relating amount of drug in body to plasma concentration. Not a real volume — a proportionality constant. Units: L or L/kg. Small Vd (~3 L): drug confined to plasma (highly protein bound — Warfarin). Large Vd (>>42 L): drug sequestered in tissues (Chloroquine ~13,000 L). Clinical: large Vd drugs cannot be efficiently removed by dialysis. Factors affecting Vd: protein binding (↑PB → ↓Vd), tissue binding (↑tissue binding → ↑Vd), body composition (↑fat → ↑Vd for lipophilic drugs).
- 10Area Under the Curve (AUC): AUC = total drug exposure over time (mg·h/L or μg·h/mL). Calculated by: (1) Trapezoidal rule: AUC = Σ [(Cp_n + Cp_n+1)/2 × Δt]. (2) For IV bolus: AUC₀₋∞ = C₀/KE. (3) For oral: AUC₀₋∞ = F·Dose/CL. AUC reflects EXTENT of absorption (bioavailability). Used in: bioequivalence studies (compare AUC_test vs AUC_ref), clearance calculation (CL = F·Dose/AUC), dose proportionality studies.
- 11Absorption Rate Constant (Ka): First-order rate constant for drug absorption from administration site. Units: h⁻¹. Determined by: Method of Residuals (feathering) from oral Cp-time data. Steps: (1) Plot ln(Cp) vs time. (2) Extrapolate terminal straight line (slope = −KE) back. (3) Residual = extrapolated value − observed value during absorption phase. (4) Plot ln(residual) vs time → slope = −Ka. Ka > KE for typical absorption (flip-flop kinetics occurs when Ka < KE → terminal slope reflects Ka, not KE — in sustained-release formulations).
- 12Clearance (CL): CL = rate of drug elimination / plasma concentration. Represents the volume of plasma completely cleared of drug per unit time (L/h or mL/min). Total body clearance: CLt = Dose/AUC (IV bolus), CLt = F·Dose/AUC (oral). CLt = CLR + CLNR (renal + non-renal). CLR = fe × CLt (where fe = fraction excreted unchanged in urine). Organ clearance: CLorgan = Q × E (Q = blood flow, E = extraction ratio). High extraction ratio drugs (E > 0.7): CL depends on blood flow (flow-limited — Lidocaine, Morphine). Low extraction ratio (E < 0.3): CL depends on protein binding and intrinsic clearance (capacity-limited — Warfarin, Phenytoin).
Learning Objectives
Exam Prep Questions
Q1. Why does it take 4–5 half-lives to reach steady state?
At steady state, rate of drug input = rate of drug elimination. Mathematically: Css = (R₀/CL) × (1 − e^(-KE·t)). After 1 t½: 50% of Css reached. After 2 t½: 75%. After 3 t½: 87.5%. After 4 t½: 93.75%. After 5 t½: 96.875% (≈97%). This is independent of the infusion rate — doubling the rate doubles the final Css but doesn’t change the TIME to reach it. Only changing t½ (by changing KE or Vd) affects the time to reach steady state.
Q2. What is Flip-Flop Kinetics?
Normally Ka >> KE (absorption is much faster than elimination) → the terminal slope of the oral Cp-time curve reflects KE. In flip-flop kinetics: Ka << KE (absorption is SLOWER than elimination — as in sustained-release formulations or poorly absorbed drugs). The terminal slope then reflects Ka (not KE), and KE is obtained from the residual line. The PK parameters ‘flip’ — the roles of Ka and KE reverse in the equations. This is identified by comparing the terminal slope from oral data with the KE from IV data.
Q3. What is the difference between CL and KE?
KE (elimination rate constant, h⁻¹): fraction of drug eliminated per unit time. First-order rate constant. CL (clearance, L/h): volume of plasma completely cleared of drug per unit time. CL has a physical meaning (volume cleared), while KE is a rate constant. Relationship: CL = KE × Vd. Clinical preference: CL is more useful because it directly relates to dosing — at steady state: Dose/τ = CL × Css. CL is generally constant (organ function), while KE may vary if Vd changes (disease, aging, obesity).